Monday, April 25, 2016

Gruesome Grammar

A week or so back we observed the rough consensus that basic arithmetic operations are essentially some kind of prepositions. Coincidentally, tonight I'm reviewing the current edition of "CK-12 Algebra - Basic" (Kramer, Gloag, Gloag; May 30, 2015) -- and the very first thing in the book is to get this exactly wrong. Here are the first two paragraphs in the book (Sec 1.1):

When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.

A verb is a “doing” word, such as running, jumps, or drives. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.

That's the kind of thing that consistently aggravates me about open-source textbooks. While I love and agree with establishing algebraic notation as a kind of language -- as possibly the single most important overriding concept of the course -- to get the situation exactly wrong right off the bat (as well as using descriptors like PEMDAS, god help us) keep these as fundamentally unusable in my courses. This, of course, then leaves no grammatical position at all for relations (like equals) when they finally appear later in the text (Sec. 1.4). So close, and yet so far.

4 comments:

I agree wholeheartedly that mathematics is a language unto itself. Perhaps it would be better to say that it has several languages, as the language of arithmetic is quite a bit different from that of, say, graph theory. However, we may be picking nits in trying to label the "parts of speech."

Consider "three plus five" versus "add three to five." In the former, "plus" is a preposition; in the latter, "add" is a verb. But the two statements clearly refer to the same operation: addition. They are two ways of saying the same thing.

Perhaps the mathematical parts of speech don't map cleanly onto those of English. I can see where the author of the book you cited may be coming from: operators are different from functions only in notation. Otherwise, they do the same thing. The arithmetic operators are functions that consume two inputs and produce an output. In this view, all functions (and by extension, all operators) are verbs: they are an action applied to nouns regardless of how that action is actually expressed.

I'm curious, though. What part of speech is a relation? Is it an assertive statement ("x is less than four") or a predicate ("is x less than four?")?

I'm very sympathetic to this, in that the first time I thought about it, I wanted operations to be verbs, and numbers to be nouns. But the truth is that algebraic writing doesn't have commands or questions or notion of an understood "you"; only assertions (and you need to wrap it in English to actually make any question).

The expression 3 + 5, equivalent to "three plus five" is not a full statement, but only a sentence fragment. To make a full English sentence you need a relation symbol, like 3 + 5 = 8, indicating that it's relations that are truly the verbs. In particular, equality is equivalent to "is" or "to be" the most basic verb in any language.

At least that's what seems clear to me. I do have one fairly old Introductory College Algebra text (Rietz and Crathorne, 1923) that asserts an equation is "an interrogative sentence, asking for values of the letters that make the members equal. Thus, x - 5 = 6 asks for the value of x that makes x - 5 = 6" (article 19). This weirds me out immensely, and I haven't seen any interpretation like that in more modern books. Maybe it's my CS background but it seems much easier to interpret equations as declarative sentences (equals as the verb "is") that can be evaluated either true (like 3 + 5 = 8) or false (like 0 = 1).

And this further highlights that operations can't be the verbs, but rather preposition-style modifiers.